Sets, Relations & Functions · JEE Main previous year questions

2023 67 questions

1. Q72023 · 29 Jan 2023 · Shift 2

   Consider a function $f:\mathbb{N}\to \mathbb{R}$, satisfying $f(1)+2f(2)+3f(3)+\dots +xf(x)=x(x+1)f(x);x\ge 2$ with $f(1)=1$. Then $\frac{1}{f(2022)}+\frac{1}{f(2028)}$ is equal to

   1. A.8000
   2. B.8400
   3. C.8100
   4. D.8200
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   Answer: C

2. Q82023 · 29 Jan 2023 · Shift 2

   The set of all values of $\lambda$ for which the equation ${\cos }^{2}2x-2{\sin }^{4}x-2{\cos }^{2}x=\lambda$ has a real solution $x$, is :

   1. A.$[-2,-1]$
   2. B.$\left[-\frac{3}{2},-1\right]$
   3. C.$\left[-2,-\frac{3}{2}\right]$
   4. D.$\left[-1,-\frac{1}{2}\right]$
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   Answer: B

3. Q152023 · 29 Jan 2023 · Shift 2

   Let $f$ and $g$ be the twice differentiable functions on $\mathbb{R}$ such that $f^{\prime \prime }(x)=g^{\prime \prime }(x)+6x$ $f^{\prime }(1)=4g^{\prime }(1)-3=9$ $f(2)=3g(2)=12$. Then which of the following is NOT true?

   1. A.$g(-2)-f(-2)=20$
   2. B.There exists $x_0\in (1,3/2)$ such that $f(x_0)=g(x_0)$
   3. C.$|f^{\prime }(x)-g^{\prime }(x)|<6\Rightarrow -1<x<1$
   4. D.If $-1<x<2$, then $|f(x)-g(x)|<8$
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   Answer: D

4. Q22023 · 25 Jan 2023 · Shift 1

   The minimum value of the function $f(x)={\int }_0^2{e}^{|x-t|}dt$ is :

   1. A.$2$
   2. B.$2(e-1)$
   3. C.$e(e-1)$
   4. D.$2e-1$
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   Answer: B

5. Q32023 · 25 Jan 2023 · Shift 1

   Let $x=2$ be a local minima of the function $f(x)=2x^4-18x^2+8x+12$, $x\in (-4,4)$. If $\mathrm{M}$ is local maximum value of the function $f$ in $(-4,4)$, then $\mathrm{M}=$

   1. A.$18\sqrt{6}-\frac{33}{2}$
   2. B.$18\sqrt{6}-\frac{31}{2}$
   3. C.$12\sqrt{6}-\frac{33}{2}$
   4. D.$12\sqrt{6}-\frac{31}{2}$
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   Answer: C

6. Q122023 · 25 Jan 2023 · Shift 1

   Let $z_1=2+3i$ and $z_2=3+4i$. The set $\mathrm{S}=\left\{z\in \mathbb{C}:|z-z_1{|}^2-|z-z_2{|}^2=|z_1-z_2{|}^2\right\}$ represents a

   1. A.hyperbola with the length of the transverse axis 7
   2. B.hyperbola with eccentricity 2
   3. C.straight line with the sum of its intercepts on the coordinate axes equals $-18$
   4. D.straight line with the sum of its intercepts on the coordinate axes equals 14
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   Answer: D

7. Q152023 · 25 Jan 2023 · Shift 1

   Let $f:(0,1)\to \mathbb{R}$ be a function defined $f(x)=\frac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements (I) g is an increasing function in $(0,1)$ (II) g is one-one in $(0,1)$ Then,

   1. A.Both (I) and (II) are true
   2. B.Neither (I) nor (II) is true
   3. C.Only (II) is true
   4. D.Only (I) is true
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   Answer: A

8. Q222023 · 25 Jan 2023 · Shift 1· Numerical answer

   Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is _____________.

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   Answer: 43

9. Q242023 · 25 Jan 2023 · Shift 1· Numerical answer

   For some $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathbb{N}$, let $f(x)=ax-3$ and $\mathrm{g}(x)=x^{\mathrm{b}}+\mathrm{c},x\in \mathbb{R}$. If $(fog{)}^{-1}(x)={\left(\frac{x-7}{2}\right)}^{1/3}$, then $(fog)(\mathrm{ac})+(gof)(\mathrm{b})$ is equal to ____.

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   Answer: 2039

10. Q32023 · 25 Jan 2023 · Shift 2

    The number of functions $f:\{1,2,3,4\}\to \{a\in \mathbb{Z}:|a|\le 8\}$ satisfying $f(n)+\frac{1}{n}f(n+1)=1,\forall n\in \{1,2,3\}$ is

    1. A.2
    2. B.3
    3. C.1
    4. D.4
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    Answer: A

11. Q92023 · 25 Jan 2023 · Shift 2

    Let the function $f(x)=2x^3+(2p-7)x^2+3(2p-9)x-6$ have a maxima for some value of $x<0$ and a minima for some value of $x>0$. Then, the set of all values of p is

    1. A.$\left(-\frac{9}{2},\frac{9}{2}\right)$
    2. B.$\left(\frac{9}{2},\infty \right)$
    3. C.$\left(0,\frac{9}{2}\right)$
    4. D.$\left(-\infty ,\frac{9}{2}\right)$
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    Answer: D

12. Q122023 · 25 Jan 2023 · Shift 2

    Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)={\log }_{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}$, for some $m$, such that the range of $f$ is [0, 2]. Then the value of $m$ is ____

    1. A.4
    2. B.3
    3. C.5
    4. D.2
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    Answer: C

13. Q32023 · 24 Jan 2023 · Shift 1

    The relation $R=\{(a,b):\gcd (a,b)=1,2a\ne b,a,b\in \mathbb{Z}\}$ is :

    1. A.reflexive but not symmetric
    2. B.transitive but not reflexive
    3. C.symmetric but not transitive
    4. D.neither symmetric nor transitive
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    Answer: D

14. Q92023 · 24 Jan 2023 · Shift 1

    The equation $x^2-4x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has :

    1. A.exactly two solutions in $(-\infty ,\infty )$
    2. B.no solution
    3. C.a unique solution in $(-\infty ,\infty )$
    4. D.a unique solution in $(-\infty ,1)$
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    Answer: C

15. Q212023 · 24 Jan 2023 · Shift 1· Numerical answer

    Let $\lambda \in \mathbb{R}$ and let the equation E be $|x{|}^2-2|x|+|\lambda -3|=0$. Then the largest element in the set $S=\{x+\lambda :x\text{ is an integer solution of E}\}$ is ____

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    Answer: 5

16. Q12023 · 24 Jan 2023 · Shift 2

    Let $f(x)$ be a function such that $f(x+y)=f(x).f(y)$ for all $x,y\in \mathbb{N}$. If $f(1)=3$ and ${\sum }_{k=1}^{n}f(k)=3279$, then the value of n is

    1. A.9
    2. B.7
    3. C.6
    4. D.8
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    Answer: B

17. Q212023 · 24 Jan 2023 · Shift 2· Numerical answer

    The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation, is __________.

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    Answer: 13